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This chapter approaches aspects of unstable homotopy theory from the perspective of the preceding chapter: We study the unstable analog of effective descent and thereby embark on an analysis of the algebro–geometric objects that arise from these considerations. This lends a modern perspective on a variety of classical topics: Hopf rings, the Ravenel–Wilson results on the infinite loopspaces associated to BP, Dieudonné theory, Brown–Gitler spectra, and the formal schemes associated to Eilenberg–Mac Lane spaces. We also include a novel discussion of the functorial algebraic geometry of Hopf rings, which gives a reinterpretation of the central Ravenel–Wilson relation and exerts an overall influence on our presentation of the other materials.

Let A$^′$be a complete characteristic (0,p) discrete valuation ring with absolute ramification degree e and a perfect residue field. We are interested in studying the category${\mathcal F}$${\mathcal F}$$_A$$_′$of finite flat commutative group schemes over A$^′$withp-power order. When e= 1, Fontaine formulated the purely ’linear algebra‘ notion of a finite Honda system over A$^′$and constructed an anti-equivalence of categories between${\mathcal F}$${\mathcal F}$$_A$$_′$and the category of finite Honda systems over A$^′$ when p< 2. We generalize this theory to the case e≤ − 1.

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4 - Unstable Cooperations

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Finite Group Schemes over Bases with Low Ramification

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2 results

4 - Unstable Cooperations

Summary

Finite Group Schemes over Bases with Low Ramification